Optimal. Leaf size=75 \[ \frac{2 a}{d e \sqrt{e \cot (c+d x)}}-\frac{\sqrt{2} a \tanh ^{-1}\left (\frac{\sqrt{e} \cot (c+d x)+\sqrt{e}}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{d e^{3/2}} \]
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Rubi [A] time = 0.0858923, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3529, 3532, 208} \[ \frac{2 a}{d e \sqrt{e \cot (c+d x)}}-\frac{\sqrt{2} a \tanh ^{-1}\left (\frac{\sqrt{e} \cot (c+d x)+\sqrt{e}}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{d e^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3529
Rule 3532
Rule 208
Rubi steps
\begin{align*} \int \frac{a+a \cot (c+d x)}{(e \cot (c+d x))^{3/2}} \, dx &=\frac{2 a}{d e \sqrt{e \cot (c+d x)}}+\frac{\int \frac{a e-a e \cot (c+d x)}{\sqrt{e \cot (c+d x)}} \, dx}{e^2}\\ &=\frac{2 a}{d e \sqrt{e \cot (c+d x)}}-\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{2 a^2 e^2-e x^2} \, dx,x,\frac{a e+a e \cot (c+d x)}{\sqrt{e \cot (c+d x)}}\right )}{d}\\ &=-\frac{\sqrt{2} a \tanh ^{-1}\left (\frac{\sqrt{e}+\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{d e^{3/2}}+\frac{2 a}{d e \sqrt{e \cot (c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.242777, size = 191, normalized size = 2.55 \[ \frac{a \left (8 \tan ^{\frac{3}{2}}(c+d x) \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},-\tan ^2(c+d x)\right )+6 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )-6 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )+24 \sqrt{\tan (c+d x)}+3 \sqrt{2} \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )-3 \sqrt{2} \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )\right )}{12 d \tan ^{\frac{3}{2}}(c+d x) (e \cot (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.021, size = 355, normalized size = 4.7 \begin{align*} -{\frac{a\sqrt{2}}{4\,d{e}^{2}}\sqrt [4]{{e}^{2}}\ln \left ({ \left ( e\cot \left ( dx+c \right ) +\sqrt [4]{{e}^{2}}\sqrt{e\cot \left ( dx+c \right ) }\sqrt{2}+\sqrt{{e}^{2}} \right ) \left ( e\cot \left ( dx+c \right ) -\sqrt [4]{{e}^{2}}\sqrt{e\cot \left ( dx+c \right ) }\sqrt{2}+\sqrt{{e}^{2}} \right ) ^{-1}} \right ) }-{\frac{a\sqrt{2}}{2\,d{e}^{2}}\sqrt [4]{{e}^{2}}\arctan \left ({\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ) }+{\frac{a\sqrt{2}}{2\,d{e}^{2}}\sqrt [4]{{e}^{2}}\arctan \left ( -{\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ) }+{\frac{a\sqrt{2}}{4\,de}\ln \left ({ \left ( e\cot \left ( dx+c \right ) -\sqrt [4]{{e}^{2}}\sqrt{e\cot \left ( dx+c \right ) }\sqrt{2}+\sqrt{{e}^{2}} \right ) \left ( e\cot \left ( dx+c \right ) +\sqrt [4]{{e}^{2}}\sqrt{e\cot \left ( dx+c \right ) }\sqrt{2}+\sqrt{{e}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{e}^{2}}}}}+{\frac{a\sqrt{2}}{2\,de}\arctan \left ({\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{e}^{2}}}}}-{\frac{a\sqrt{2}}{2\,de}\arctan \left ( -{\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{e}^{2}}}}}+2\,{\frac{a}{de\sqrt{e\cot \left ( dx+c \right ) }}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.00845, size = 803, normalized size = 10.71 \begin{align*} \left [\frac{4 \, a \sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} \sin \left (2 \, d x + 2 \, c\right ) + \frac{\sqrt{2}{\left (a e \cos \left (2 \, d x + 2 \, c\right ) + a e\right )} \log \left (\frac{\sqrt{2} \sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}{\left (\cos \left (2 \, d x + 2 \, c\right ) - \sin \left (2 \, d x + 2 \, c\right ) - 1\right )}}{\sqrt{e}} + 2 \, \sin \left (2 \, d x + 2 \, c\right ) + 1\right )}{\sqrt{e}}}{2 \,{\left (d e^{2} \cos \left (2 \, d x + 2 \, c\right ) + d e^{2}\right )}}, \frac{\sqrt{2}{\left (a e \cos \left (2 \, d x + 2 \, c\right ) + a e\right )} \sqrt{-\frac{1}{e}} \arctan \left (\frac{\sqrt{2} \sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} \sqrt{-\frac{1}{e}}{\left (\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) + 1\right )}}{2 \,{\left (\cos \left (2 \, d x + 2 \, c\right ) + 1\right )}}\right ) + 2 \, a \sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} \sin \left (2 \, d x + 2 \, c\right )}{d e^{2} \cos \left (2 \, d x + 2 \, c\right ) + d e^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} a \left (\int \frac{1}{\left (e \cot{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, dx + \int \frac{\cot{\left (c + d x \right )}}{\left (e \cot{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a \cot \left (d x + c\right ) + a}{\left (e \cot \left (d x + c\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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